Cryptography Reference
In-Depth Information
closest to
is often called the
quanta
.Expo-
nential or logarithmic schemes may be appropriate for some
cases.
x
.Thevalueof
Q
5. Let
be a watermark. Insert a watermark by tweaking
each coefficient. Each value of
{b
0
,...,b
n
}
c
i
lies between an odd and an
even integer multiple of
Q
.Thatis,
k
i
Q ≤ c
i
≤
(
k
i
+1)
Q
.To
encode
b
i
=0
at coefficient
c
i
,set
c
i
to the even multiple of
Q
.
To encode
b
i
=1
,set
c
i
to be the odd multiple of
Q
.
6. Use a reverse transform to reconstruct the original document
from the new values of
{c
0
,...,c
n
}
.Iftheunderstandingofthe
decomposition process is correct, the changes will not alter the
image dramatically. Of course, some experimentation with the
value of
Q
may be necessary.
This scheme for encoding a watermark can be used with many
models for deconstructing images and sound files. The greatest chal-
lenge is setting the value of
adds robustness
at the cost of introducing greater distortion. The cost of a large
Q
correctly. A large
Q
Q
should be apparent by this point. The value can be seen by examin-
ing the algorithm for extracting the watermark:
1. To extract a watermark, begin by applying the same decon-
structive technique that models the document as a series of co-
efficients:
{c
0
,...,c
n
}
.
|k
i
Q − c
i
|
2. Find the integer
k
i
such that
is minimized. If there's
c
i
=
been no distortion in the image, then
.Ifourmodelof
the document is good, small changes in the document should
correspond to small changes in the coefficients. Small changes
should still result in the same values of
k
i
Q
k
i
.
3. If
k
i
is odd, then
b
i
=1
.If
k
i
is even, then
b
i
=0
. This is the
watermark.
Many watermarking schemes add an additional layer of protec-
tion by including some error correction bits to the watermark bits,
{b
0
,...,b
n
}
. (See Chapter 3.) Another solution is to compare the
watermarking bits to a known set of watermarks and find the best
match. To some extent, these are the same techniques. Choosing the
size of
and the amount of error correction lets you determine the
amount of robustness available.
This solution is a very good approach that relies heavily on the
decomposition algorithm. The discrete cosine transform is a good
solution, but it has weaknesses. Even slight rotations can introduce
Q
